Synthetic Priors
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912932195991552 |
|---|---|
| author | Polson, Nick Sokolov, Vadim |
| author_facet | Polson, Nick Sokolov, Vadim |
| contents | Bayesian inference in generalized linear models requires a prior on the coefficient vector $β$. Practitioners naturally reason about response probabilities at specific covariate values, not about abstract log-odds parameters. We develop synthetic priors: informative Bayesian priors for GLMs grounded in Good's device of imaginary observations -- the principle that every conjugate prior is equivalent to a likelihood on pseudo-data from the same exponential family. The conditional means prior of Bedrick (1996) elicits independent Beta priors on the conditional mean response at $p$ expert-chosen design points; the induced prior on $β$ is a product of binomial likelihoods at synthetic data points. Combined with Pólya-Gamma data augmentation \citep{polson2013}, the posterior admits an exact conjugate Gibbs sampler -- no tuning, no Metropolis step -- by treating the augmented dataset as a standard logistic regression. We show that ridge regression and catalytic priors \citep{huang2020} are instances of Good's device, and identify prediction-powered inference \citep{angelopoulos2023ppi} as a structural analogue in the frequentist setting -- all three mediate a variance-bias tradeoff through a single informativeness parameter. We illustrate the approach on two benchmark problems: the Challenger O-ring data \citep{dalal1989}, where the BCJ prior provides a more moderate posterior predictive at the 31°F launch temperature; and a Phase~II atopic dermatitis dose-finding trial ($n = 300$), where the synthetic prior narrows 95\% credible intervals by 3-6\% and raises decision probabilities by up to 2 percentage points relative to a flat prior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00347 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Synthetic Priors Polson, Nick Sokolov, Vadim Methodology Bayesian inference in generalized linear models requires a prior on the coefficient vector $β$. Practitioners naturally reason about response probabilities at specific covariate values, not about abstract log-odds parameters. We develop synthetic priors: informative Bayesian priors for GLMs grounded in Good's device of imaginary observations -- the principle that every conjugate prior is equivalent to a likelihood on pseudo-data from the same exponential family. The conditional means prior of Bedrick (1996) elicits independent Beta priors on the conditional mean response at $p$ expert-chosen design points; the induced prior on $β$ is a product of binomial likelihoods at synthetic data points. Combined with Pólya-Gamma data augmentation \citep{polson2013}, the posterior admits an exact conjugate Gibbs sampler -- no tuning, no Metropolis step -- by treating the augmented dataset as a standard logistic regression. We show that ridge regression and catalytic priors \citep{huang2020} are instances of Good's device, and identify prediction-powered inference \citep{angelopoulos2023ppi} as a structural analogue in the frequentist setting -- all three mediate a variance-bias tradeoff through a single informativeness parameter. We illustrate the approach on two benchmark problems: the Challenger O-ring data \citep{dalal1989}, where the BCJ prior provides a more moderate posterior predictive at the 31°F launch temperature; and a Phase~II atopic dermatitis dose-finding trial ($n = 300$), where the synthetic prior narrows 95\% credible intervals by 3-6\% and raises decision probabilities by up to 2 percentage points relative to a flat prior. |
| title | Synthetic Priors |
| topic | Methodology |
| url | https://arxiv.org/abs/2603.00347 |